Answer to Question #120643 in Algebra for anav

Question #120643
A quadratic equation f (x) has two roots α and β . If α=-4+5i, determine the root β
and thus obtain an expression for the equation f (x)
1
Expert's answer
2020-06-08T19:29:55-0400

If α=4+5i\alpha=-4+5i is the root of the quadratic equation f(x)=0,f(x)=0, then β=45i\beta=-4-5i is also the root of the quadratic equation f(x)=0.f(x)=0.

α=4+5i,β=45i\alpha=-4+5i, \beta=-4-5i

Hence


f(x)=a(xα)(xβ),a0f(x)=a(x-\alpha)(x-\beta), a\not=0

f(x)=a(x(4+5i))(x(45i))f(x)=a(x-(-4+5i))(x-(-4-5i))

α+β=4+5i45i=8=ba\alpha+\beta=-4+5i-4-5i=-8=-{b\over a}

αβ=(4+5i)(45i)=(4)2+(5)2=41=ca\alpha\cdot\beta=(-4+5i)\cdot(-4-5i)=(-4)^2+(5)^2=41={c\over a}

f(x)=ax2+8ax+41a,a0f(x)=ax^2 +8ax+41a, a\not=0


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